NYSTCE Math CST cheat sheet

Polya's four phases: understand the problem, devise a plan, carry out the plan, look back and verify-each phase has a checkable output.

Work backward when the final result is given and operations are reversible; work forward when initial conditions are given and the goal is a single output.

A counterexample disproves a universal claim; one confirming example never proves it. Proof requires a general argument covering all cases.

Inductive reasoning generalizes from observed cases and yields conjectures; deductive reasoning derives conclusions from accepted premises and yields proofs.

Estimation check: round each quantity to one significant figure and confirm the computed answer's order of magnitude matches the estimate.

Dimensional analysis: treat units as algebraic factors, cancel them across multiplication and division, and require both sides of an equation to share units.

Modeling cycle: identify variables, formulate relationships, compute, interpret results in context, then validate and revise the model.

When a problem states a rate, write it as a ratio with units (e.g., miles per hour) so conversion factors can be applied by unit cancellation.

Real number hierarchy: naturals ⊂ integers ⊂ rationals ⊂ reals; rationals are terminating or repeating decimals, irrationals are neither.

Divisibility: by 3 if digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 9 if digit sum is divisible by 9; by 6 if divisible by both 2 and 3.

GCD via prime factorization: multiply the lowest power of each shared prime; LCM: multiply the highest power of every prime appearing.

Rules of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m−n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^(1/n) = nth root of a.

Complex numbers: i² = −1; (a+bi)(c+di) = (ac−bd) + (ad+bc)i; divide by multiplying numerator and denominator by the conjugate of the denominator.

Percent change = (new − old)/old × 100; successive percent changes multiply, so a 20% increase then 20% decrease yields a net 4% decrease.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts, and raising to a power multiplies the exponent.

Absolute value |x − a| equals the distance from x to a on the number line; |x − a| < d means a − d < x < a + d.

Slope m = (y₂ − y₁)/(x₂ − x₁); point-slope form y − y₁ = m(x − x₁); parallel lines share slope, perpendicular slopes multiply to −1.

Quadratic formula x = (−b ± √(b² − 4ac))/(2a); discriminant b² − 4ac > 0 gives two real roots, = 0 one repeated root, < 0 two complex roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the parabola opens upward when a > 0 and downward when a < 0.

Exponential growth/decay y = a·b^t: b > 1 growth, 0 < b < 1 decay; compound interest A = P(1 + r/n)^(nt); continuous growth A = Pe^(rt).

Function composition (f∘g)(x) = f(g(x)) is generally not commutative; the inverse f⁻¹ exists only if f is one-to-one, and graphs of inverses reflect across y = x.

Logarithm rules: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^k) = k log x; log_b x = ln x / ln b for any base b.

Transformations of f(x): f(x − h) shifts right h, f(x) + k shifts up k, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis, a·f(x) scales vertically.

Systems: a unique solution when slopes differ, no solution when slopes are equal and intercepts differ, infinitely many when the equations are equivalent.

Angle relations: vertical angles are equal; angles on a straight line sum to 180°; interior angles of any n-gon sum to (n − 2)·180°.

Similar figures have equal corresponding angles and proportional sides; areas scale by the square of the ratio and volumes by the cube of the ratio.

Circle measures: circumference 2πr, area πr², arc length (θ/360)·2πr, sector area (θ/360)·πr², with θ in degrees.

Volume formulas: cylinder πr²h, cone (1/3)πr²h, sphere (4/3)πr³, prism Bh; surface area of a sphere is 4πr².

Distance and midpoint: d = √((x₂−x₁)² + (y₂−y₁)²); midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

Pythagorean theorem a² + b² = c²; common triples 3-4-5, 5-12-13, 8-15-17; 45-45-90 sides are x, x, x√2 and 30-60-90 sides are x, x√3, 2x.

Right-triangle ratios: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent; law of sines a/sin A = b/sin B; law of cosines c² = a² + b² − 2ab·cos C.

Coordinate geometry: area of a polygon from vertex coordinates via the shoelace formula; a line perpendicular to one with slope m has slope −1/m.

Mean = sum/n; median = middle value of ordered data; mode = most frequent value; the mean is pulled toward skew and outliers while the median resists them.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves it unchanged, multiplying every value by k multiplies it by |k|.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B); P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

nPr = n!/(n − r)! counts ordered arrangements; nCr = n!/(r!(n − r)!) counts unordered selections; with replacement, counts multiply per trial.

Normal distribution: about 68% of data within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3; convert to z = (x − μ)/σ.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not establish causation.

Line of best fit predicts y from x; the residual is observed minus predicted, and a residual plot showing a pattern indicates a poor linear fit.

Margin of error decreases as sample size grows and as confidence level falls; a random sample is what licenses generalization to the population.